A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules

A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules Michael B. Gordy(cid:3) Board ofGovernorsoftheFederal ReserveSystem October22,2002 Abstract When economic capital is calculated using a portfolio model of credit value-at-risk, the marginal capital requirement for an instrument depends, in general, on the properties of the portfolio in which it is held. Bycontrast, ratings-based capital rules, including both the current Basel Accord and itsproposed revision, assign a capital charge to an instrument based only on its own characteristics. I demonstrate that ratingsbasedcapital rulescanbereconciled withthegeneral classofcredit VaRmodels. Contributions toVaRare portfolio-invariant only if(a) there isonly a single systematic risk factor driving correlations across obligors, and (b) no exposure in a portfolio accounts for more than an arbitrarily small share of total exposure. Analysis of rates of convergence to asymptotic VaR leads to a simple and accurate portfolio-level add-on charge for undiversified idiosyncratic risk. There is no similarly simple way to address violation of the singlefactorassumption. JELCodes: G31,G38 (cid:3) Theviewsexpressedhereinaremyownanddonotnecessarily reflectthoseoftheBoardofGovernors oritsstaff. IwouldliketothankPaulCalem,DarrellDuffie,PaulEmbrechts,JonFaust,ErikHeitfield,David Jones,DavidLando,GennadySamorodnistky, DirkTasche,andTomWildefortheirhelpfulcomments,and Susan Yeh for editorial suggestions. Please address correspondence to the author at Division of Research andStatistics,MailStop153,FederalReserveBoard,Washington,DC20551,USA.Phone: (202)452-3705. Fax: (202)452-5295. Email: hmichael.gordy@frb.govi.

Recentyearshavewitnessedsignificant advancesinthedesign, calibration andimplementation ofportfolio models of credit risk. Large commercial banks and other financial institutions with significant credit exposure rely increasingly on models to guide credit risk management at the portfolio level. Models allow management toidentify concentrations ofriskandopportunities fordiversification withinadisciplined and objective framework, and thus offer a more sophisticated, less arbitrary alternative to traditional lending limitcontrols. Morewidespreadandintensiveuseofmodelsisencouraging amoreactiveapproachtoportfoliomanagementatcommercialbanks,whichhascontributedtotheimprovedliquidityofmarketsfordebt instruments andcreditderivatives. Strippedtoitsessentials,acreditriskmodelisafunctionmappingfromaparsimonioussetofinstrumentlevelcharacteristics andmarket-level parameters toadistribution forportfolio credit losses oversomechosenhorizon. Themodeloutputofprimaryinterest,the“economiccapital”requiredtosupporttheportfolio, is derived as some summary statistic of the loss distribution. The definition of economic capital in most widespread use is value-at-risk (“VaR”). Under the VaR paradigm, an institution holds capital in order to maintain a target rating for its own debt. Associated with the target rating is a probability of survival over thehorizon (say,99.9%overoneyear). Tobeconsistent withitstargetsurvival probability (denoted q),the th institution musthold reserves andequity capital sufficient tocoveruptotheq quantile ofthedistribution ofportfolio lossoverthehorizon. Directlyorindirectly, modelapplications toactiveportfoliomanagement depend onthecapacity tomeasure howtheportfolio capital requirement changes withchanges inportfolio composition. From a public policy perspective, model-based measurement of economic capital offers a potentially attractive solution to an increasingly urgent regulatory problem. The current regulatory framework for required capital on commercial bank lending is based on the 1988 Basel Accord. Under the Accord, the capital charge on commercial loans is a uniform 8% of loan face value, regardless of the financial strength oftheborrowerorthequalityofcollateral.1 TheAccord’srulesarerisk-sensitive onlyinthatlowercharges are specified for certain special classes of lending, e.g., to OECD member governments, to other banks in OECD countries, or for residential mortgages. When the Accord was first introduced, the 8% charge appeared to be “about right on average” for a typical bank portfolio. Over time, however, the failure to distinguish among commercial loans of very different degrees of credit risk created the incentive to move low-riskinstrumentsoffbalancesheetandretainonlyrelativelyhigh-riskinstruments. Thefinancialinnovationswhicharoseinresponse tothisincentivehaveundermined theeffectiveness ofregulatory capitalrules (see,e.g.,Jones2000)andthusledtocurrenteffortstowardsreform. Itiswidelyrecognizedthatregulatory arbitrage willcontinue untilregulatory capitalchargesattheinstrumentlevelarealignedmorecloselywith underlying risk. The Basel Committee on Bank Supervision (1999) undertook a detailed study of how banks’ internal modelsmightbeusedforsettingregulatorycapital. TheCommitteeacknowledgedthatacarefullyspecified and calibrated model could deliver a more accurate measure of portfolio credit risk than any rule-based system, but found that the present state of model development could not ensure an acceptable degree of 1Theso-called8%ruletakesaratherbroaddefinitionof“capital.” Ineffect,roughlyhalfthis8%mustbeinequitycapital,as measuredonabook-valuebasis. 1

comparabilityacrossinstitutionsandthatdataconstraintswouldpreventvalidationofkeymodelparameters andassumptions.2 Itseemsunlikely, therefore, thatregulatorswillbepreparedinthenear-tomedium-term to accept the use of internal models for setting regulatory capital. Nonetheless, regulators and industry practitioners appear to be in broad agreement that a revised Accord should permit evolution towards an internalmodelsapproach asmodelsanddataimprove. Atpresent,itappearsvirtuallycertainthatareformedAccordwillofferaratings-based“risk-bucketing” systemofoneformoranother. Insuchasystem,bankingbookassetsaregroupedinto“buckets,”whichare presumed tobehomogeneous. Associated witheachbucketisafixedcapitalcharge perdollar ofexposure. InthelatestversionoftheBaselproposalforanInternalRatings-Based(“IRB”)approach(BaselCommittee on Bank Supervision 2001), the bucketing system is required to partition instruments by internal borrower rating; by loan type (e.g., sovereign vs. corporate vs. project finance); by one or more proxies for seniority/collateral type, which determines loss severity in the event of default; and by maturity. More complex systems might further partition instruments by, for example, country and industry of borrower. Regardless ofthesophistication ofthebucketing scheme,capitalcharges areportfolio-invariant, i.e.,thecapitalcharge on a given instrument depends only on its own characteristics, and not the characteristics of the portfolio in which it is held. I take portfolio-invariance to be the essential property of ratings-based capital rules. Throughoutthispaper,Iwillusetheterm“ratings-based” toreferbroadlytoportfolio-invariant capitalallocationruleswithbucketingalongmultipledimensions,ratherthantoconstrainthetermtoschemesinwhich capitaldepends onlyonatraditional univariate creditrating. A regulatory regime based on ratings-based assignment of capital charges does offer significant advantages. The current Accord is itself a simple ratings-based framework. The proposed new Accord will introduce additional bucketing criteria and make better use of information in borrower ratings, yet still be viewed as a natural extension of the current regime. Because the capital charge for a portfolio is simply a weighted sum of the dollars in each bucket, ratings-based systems are relatively simple to administer and need not impose burdensome reporting requirements. Validation problems are also limited in scope. As the new Accord is currently envisioned, the most significant empirical challenge facing supervisors would likelyconcernthequalityofdefaultprobability estimatesforinternalgrades. Though not often recognized in the debate on regulatory reform, in practice many (if not most) large banks apply ratings-based rules for allocation of capital at the transaction level. Even at institutions that have implemented models for portfolio management and portfolio-level capital assessment, there may be reluctance to apply the implied marginal capital requirements to assess hurdle rates for individual transactions. Computational and information systems burdens may be substantial. More important perhaps, line managersarelikelytoopposeanyperformancemonitoringsysteminwhichaloanthatcouldbebookedone dayataprofitablecreditspreadbecomesunprofitablethenextdueonlytochangesinthecompositionofthe bank’s overall portfolio. Theneed forstability inbusiness operations thus favors portfolio-invariant capital charges atthetransaction level. 2Inanindustrypractitionerresponse, GARP(1999)acknowledgestheobstaclestoimmediateadoptionofaninternalmodels regulatoryregime,butarguesthatthechallengescanbemetthroughanevolutionary,piecemealapproachtoregulatorycertification ofmodelcomponents. 2

Though aratings-based scheme maybe anecessary “second-best” solution under current conditions, it is nonetheless desirable that the capital charges be calibrated within a portfolio model. Consistency with a well-specified model would bring greater discipline and accuracy to the calibration process, and would provide a smoother path of evolution towards a regime based on internal models. This paper is about the challengesinmodels-based calibrationofratings-based capitalcharges. Inparticular, itaskswhatmodeling assumptions must be imposed so that marginal contributions to portfolio economic capital are portfolioinvariant. By design, portfolio models do not, in general, yield portfolio-invariant capital charges. To obtain a distribution ofportfolioloss,amodelmustdetermineajointdistribution overcreditlossesattheinstrument level. The latest generation of widely-used models gives structure to this problem by assuming that correlations across obligors incredit events arisedue tocommon dependence onasetofsystematic risk factors. Implicitlyorexplicitly, thesefactorsrepresent thesectoralshiftsandmacroeconomicforcesthatimpingeto agreaterorlesserextentonallfirmsinaneconomy. Anaturalpropertyofthesemodelsisthatthemarginal capital required foraloandepends onhowitaffects diversification, andthusdepends onwhatelseisinthe portfolio. If economic capital is defined within the value-at-risk paradigm, then the problem has a simple answer. Ishowthattwoconditionsarenecessary and(withafewregularityconditions) sufficienttoguarantee portfolio-invariance: First, the portfolio must be asymptotically fine-grained, in the sense that no single exposure in the portfolio can account for more than an arbitrarily small share of total portfolio exposure. Second,theremustbeonlyasinglesystematic riskfactor. The emphasis in this paper is on generality across portfolios and models. The use of asymptotics to characterize model properties is not new to practitioners, but all previous analyses have been applied to homogeneous portfolios andwiththeobjective ofsimplifying computation.3 Banksvarywidelyinthesize and composition of their portfolios and in the details of their credit risk models. For policy purposes, it is essential that our results be sufficiently general to embrace this diversity. Indeed, our results are shown to applytoquiteheterogeneous portfolios andacrossabroadclassofcreditriskmodels. Needless to say, the real world does not give us perfectly fine-grained portfolios. Bank portfolios have finite numbers of obligors and lumpy distributions of exposure sizes. Capital charges calibrated to the asymptotic case, which assume that idiosyncratic risk is diversified away completely, must understate required capital for any given finite portfolio. To assess the magnitude of this bias, I determine the rate of convergence of credit value-at-risk to its asymptotic limit. As an application, I propose a simple methodology for assessing a portfolio level add-on charge to compensate for less-than-perfect diversification of idiosyncratic risk. Numerical examples suggest that the method works extremely well, so that moderate departures fromasymptotic granularity neednotposeaproblem inpractice forratings-based capitalrules. Although itisthestandard mostcommonlyapplied, value-at-risk isnotwithoutshortcomings asariskmeasurefordefiningeconomiccapital. Becauseitisbasedonasinglequantileofthelossdistribution, VaR 3Large-sampleapproximationshavebeenappliedtohomogeneousportfoliosundersingleriskfactorversionsoftheRiskMetrics Group’s CreditMetrics (Finger 1999) and KMV Portfolio Manager (Vasicek 1997) in order to obtain computational shortcuts. Bu¨rgisser, KurthandWagner(2001) characterizetheasymptoticbehavior ofageneralizedCreditRisk+ modelonasequence of portfolioswithnstatisticallyidenticalcopiesofafixedheterogeneousportfolio. 3

provides no information on the magnitude of loss incurred in the event that capital is exhausted. A more robust risk-measure is expected shortfall (“ES”), which is (loosely speaking) the expected loss conditional on being in the tail. From the perspective of an insurer of deposits (e.g., the FDIC in the US), an even more relevant risk-measure is expected excess loss (“EEL”). Under the EEL paradigm, an institution must hold enough capital so that the expected credit loss in excess ofcapital is less than or equal toatarget loss rate. Iconsider whether ESand EELdeliver portfolio-invariant capital charges for an asymptotic portfolio in a single-factor setting. Expected shortfall does, but EELdoes not, and thus is unsuitable as asoundness standard forderiving risk-bucket capitalcharges. Section 1 sets out a general framework for the class of risk-factor models in current use under a bookvaluedefinitionofcreditloss. Section2presentsthekeyresultsforVaRforthisclassofmodels. InSection 3,theseresults areshowntoapply equallytothecaseof“multi-state” modelsinwhichlossismeasuredon amarket-value basis. Acapital adjustment forundiversified idiosyncratic riskisdeveloped inSection4. In Section 5, I examine the asymptotic behavior of expected shortfall and expected excess loss as alternatives toVaR.Concludingremarksfocusontheassumptionofasinglesystematicriskfactor,whichisempirically untenableandyetanunavoidablepreconditionforportfolio-invariant capitalcharges. Whilethisassumption oughttobeacceptable inthepursuitofachievableandsubstantivenear-tomedium-termregulatoryreform, itmaylimitthelong-term viability ofratings-based risk-bucket rulesforregulatory capital. 1 A general model framework under book-value accounting Under a book-value (or actuarial) definition of loss, credit loss arises only in the event of obligor default. Change inmarket value dueto rating downgrade orupgrade isignored. Thisisthe simplest framework for our purposes, because we need only be concerned with default risk and with uncertainty in the recovery valueofanassetintheeventofobligordefault. An essential concept in any risk-factor model is the distinction between unconditional and conditional eventprobabilities. Anobligor’sunconditional defaultprobability, alsoknownasits PDorexpecteddefault frequency, istheprobability ofdefaultbeforesomehorizongivenallinformation currentlyobservable. The conditional default probability is the PD we would assign the obligor if we also knew what the realized value of the systematic risk factors at the horizon would be. The unconditional PD is the average value of the conditional default probability across all possible realizations of the systematic risk factors. Totake an example, consider a simple credit cycle in which the systematic risk factor takes only three values. The “badstate”corresponds toarecession attheriskhorizon,the“goodstate”toanexpansion, andthe“neutral state”toordinarytimes. Saythatthethreestatesoccurwithprobabilities of1=4,1=2and1=4(respectively) at the risk horizon. Consider an obligor which defaults with probability 2% in the event of a bad state, probability 1% in the neutral state, and probability 0.4% in the event of a good state. The “conditional defaultprobability”isthen0.4%,1%,or2%,dependingonwhichhorizonstateweconditionupon. ThePD istheprobability-weighted average defaultrate,or1.1%. Let X denote the systematic risk factors (possibly multivariate), which are drawn from a known joint distribution. These risk factors may be identified in some models with specific observable quantities, such 4

asmacroeconomic variablesorindustrial sectorperformance indicators, ormaybeleftabstract. Regardless of their identity, it is assumed that all correlations in credit events are due to common sensitivity to these factors. Conditional onX,theportfolio’s remaining creditriskisidiosyncratic totheindividual obligors in theportfolio. Letp (x)denotetheprobability ofdefaultforobligor iconditional onrealization xofX. i This general framework for modeling default is compatible withall ofthe best-known industry models of portfolio credit risk, including the RiskMetrics Group’s CreditMetrics, Credit Suisse Financial Prod- + uct’s CreditRisk , McKinsey’s CreditPortfolioView, and KMV’s Portfolio Manager. The similarity to + CreditRisk is easiest to see because that model is written in the language of conditional default proba- + bilities. To obtain CreditRisk within our framework, assume that the risk factors X1;::: ;X K are independent gamma-distributed random variables with mean one and variances (cid:27) 2 ;::: ;(cid:27) 2 . Let p(cid:22) denote the 1 K i PDofobligor i,andspecify p (x)as: i ! XK p (x) =p(cid:22) 1+ w (x −1) (1) i i ik k k=1 wherew isavectoroffactorloadings withsumin[0;1].4 i CreditMetrics, whichisbasedonasimplifiedMertonmodelofdefault, alsocanbecastwithinaconditional probability framework. It is assumed that the vector of risk factors X is jointly distributed N(0;Ω). Associated with each obligor is a latent variable R which represents the return on the firm’s assets. R is i i givenby R = (cid:15) −Xw ; (2) i i i i where the (cid:15) are iid N(0;1) white noise (representing obligor-specific risk) and w is a vector of factor i i loadings.5 Without loss of generality, the weights w and are scaled so that R is mean zero, variance i i i one.6 Aborrowerdefaultsifandonlyifitsassetreturnfallsbelowathreshold value γ . i To obtain the conditional default probability function p (x), observe that default occurs if and only if i (cid:15) (cid:20) (γ +Xw )= . Therefore, conditional onX=x, default byiisan independent Bernoulli eventwith i i i i probability p (x) = Pr((cid:15) (cid:20) (γ +xw )= ) = (cid:8)((γ +xw )= ) (3) i i i i i i i i where(cid:8)isthestandardnormalcdf. Tocalibratetheparameterγ ,notethattheunconditional probability of i default is(cid:8)(γ ), soγ = (cid:8)−1(p(cid:22)), wherep(cid:22) isthe PDfor obligor i.7 SeeGordy(2000) foramoredetailed i i i i derivation ofthesetwomodelsandtheirrepresentation intermsofconditional probabilities. 4Strictlyspeaking, thisfunctional form is invalid because it allows conditional probabilities to exceed one. In practice, this problemisnegligibleforhighandmoderatequalityportfoliosandreasonablecalibrationsofthe(cid:27)2. k 5TheusualwaythisisspecifiedhasXw added,notsubtracted. Thechangeinsignhereisconvenientbecauseitimpliesthat i thep (x)functionwillbeincreasinginx,butdoesnototherwisechangethestatisticalpropertiesofthemodel. i 6Specifically,theweights aregivenby(1−w0Ωw )1=2. i i i 7Byconstruction,theunconditionaldistributionofR isN(0;1),sotheprobabilitythatR (cid:20)γ is(cid:8)(γ ). i i i i 5

In some industry models, it is assumed that loss given default (“LGD”) is known and non-stochastic. Of the credit VaR models in widespread use, those that do allow for stochastic LGD always take recovery risktobepurely idiosyncratic. Inpractice, LGDnotonlymaybehighly uncertain, butmayalsobesubject to systematic risk. For example, the recovery value of defaulted commercial real estate loans depends on thevalueoftherealestatecollateral, whichislikelytobelower(higher) whenmany(few)otherrealestate projects have failed. In recent months, some progress has been made in capturing this effect. Frye (2000) develops an extension of aone-factor CreditMetrics model in which collateral values (and thus recoveries) are correlated with the same systematic risks that drive default rates. Bu¨rgisser et al. (2001) extend the + CreditRisk model to include a systematic factor for recovery risk that is orthogonal to the systematic factorsfordefaultrisk. In order to accommodate systematic and idiosyncratic recovery risk, I take loss, rather than merely default status, as the primitive outcome variable. Let A be the exposure to obligor i; these are taken to i be known and non-stochastic.8 Letthe random variable U denote loss per dollar exposure. In the event of i survival,U = 0. Otherwise,U isthepercentageLGDoninstrumenti. Theusualassumptionofconditional i i independence ofdefaults isextended toconditional independence oftheU . Iassumethat i (A-1) thefU garebounded intheunitintervaland,conditional onX,aremutuallyindependent. i Foraportfolioofnobligors,definetheportfoliolossratioL astheratiooftotallossestototalportfolio n exposure,9 i.e., P n U A L (cid:17) Pi=1 i i : (4) n n A i=1 i Foragivenq 2 (0;1),value-at-riskisdefinedastheq th percentileofthedistribution ofloss,andisdenoted VaR [L ]. Let(cid:11) (Y)denotetheq th percentile ofthedistribution ofrandom variable Y,i.e., q n q (cid:11) (Y) (cid:17) inffy :Pr(Y (cid:20) y) (cid:21) qg: (5) q Intermsofthismoregeneral notation, wehaveVaR [L ] =(cid:11) (L ). q n q n 2 Asymptotic loss distribution under book-value accounting Imaginethatthebankselectsitsportfolio asthefirstnelementsofaninfinitesequence oflending opportunities. Toguarantee that idiosyncratic risk vanishes as moreassets are added tothe portfolio, the sequence ofexposure sizesmustneitherblowupnorshrinktozerotooquickly. Iassumethat 8Inpractice,itneednotbesosimple. Iftheinstrumentisacouponbond,book-valueexposureissimplythefacevalue. Much banklending,however,isintheformoflinesofcreditwhichgivetheborrowersomecontrolovertheexposuresize.Borrowersdo tendtodrawdownunutilizedcreditlinesastheydeterioratetowardsdefault. IfweassumethatuncertaintyinAisidiosyncratic conditionalonthestateoftheobligorandisofboundedvariance,thenalltheconclusionsofthispapercontinuetohold. Inthis case,weinterpretA astheexpecteddollarexposureintheeventofobligordefault. i 9For simplicity, I assume that the portfolio contains only a single asset for each obligor. Under actuarial treatment of loss, multipleassetsofasingleobligormaybeaggregatedintoasingleassetwithoutaffectingtheresults. 6

P (A-2) the A are asequence of positive constants such that (a) n A " 1 and (b) there exists a(cid:16) > 0 i P i=1 i suchthatA = n A = O(n −(1=2+(cid:16))).10 n i=1 i The restrictions in (A-2) are sufficient to guarantee that the share of the largest single exposure in total portfolio exposure vanishes to zero as the number of exposures in the portfolio increases. As a practical matter, the restrictions are quite weak and would be satisfied by any conceivable real-world large bank portfolio. Forexample, they are satisfied ifall the A arebounded from below by apositive minimum size i andfromabovebyafinitemaximumsize. Ourfirst result isthat, under quite general conditions, the conditional distribution of L degenerates to n itsconditional expectation asn ! 1. Moreformally,wecanshowthat Proposition 1 If(A-1)and(A-2)hold,then,conditionalonX=x,L −E[L jx]! 0,almostsurely. n n The proof, which relies mainly on a strong law of large numbers, is given in Appendix A. Note that there is no restriction on the relationship between A and the distribution of U , so there is no problem if, for i i example,highqualityloanstendalsotobethelargestloans. Also,norestrictions haveyetbeenimposedon thenumberofsystematicfactorsortheirjointdistribution. In intuitive terms, Proposition 1 says that as the exposure share of each asset in the portfolio goes to zero, idiosyncratic risk in portfolio loss is diversified away perfectly. In the limit, the loss ratio converges toafixedfunction ofthesystematic factorX. Werefertothislimitingportfolio as“infinitelyfine-grained” or as an “asymptotic portfolio.” An implication is that, in the limit, we need only know the unconditional distribution ofE[L jX]toanswerquestions abouttheunconditional distribution ofL . Forexample,ifwe n n wishtoknowthevariance ofthelossratio,wecanlooktothevariance ofE[L jX]: n Proposition 2 If(A-1)and(A-2)hold,thenV[L ]−V[E[L jX]] ! 0. n n ProofisinAppendixA. A more important result is, in essence, that for any q 2 (0;1), the q th quantile of the unconditional loss distribution approaches the q th quantile of the unconditional distribution of E[L jX] as n ! 1. Our n desiredresultistohave (cid:11) (L )−(cid:11) (E[L jX]) ! 0: (6) q n q n For technical reasons, however, we are limited to a slightly restricted variant on this result. Let F denote n thecdfofL . Wecanshow: n 10FordefinitionoftheordernotationO((cid:1))seeBillingsley(1995,A18). 7

Proposition 3 If(A-1)and(A-2)hold,thenforany(cid:15) > 0 F ((cid:11) (E[L jX])+(cid:15)) ! [q;1] (7) n q n F ((cid:11) (E[L jX])−(cid:15)) ! [0;q]: (8) n q n The proof is in Appendix B. For all practical purposes, this proposition ensures that equation (6) will hold.11 The literal interpretation of Proposition 3 is that the q th quantile of E[L jX] plus an arbitrarily n small “smidgeon” (i.e., (cid:15)) is guaranteed, in the limit, to cover (or, at least, to come arbitrarily close to covering) q or more of the distribution of loss. Similarly, the q th quantile of E[L jX] less the smidgeon n th is guaranteed, in the limit, to fail to cover the q quantile of the distribution of loss (or, at least, to come arbitrarily closetosofailing). The importance of Proposition 3 is that it allows us to substitute the quantiles of E[L jX] (which typn ically are relatively easy to calculate) for the corresponding quantiles of the loss ratio L (which are hard n tocalculate) astheportfolio becomes large. Itshould beemphasized that wehave obtained thisresult with very minimal restrictions on the make-up of the portfolio and the nature of credit risk. The assets may be of quite varied PD, expected LGD,and exposure sizes. We have bounded the support of the U to the unit i interval, but have otherwise not restricted the behavior of the conditional expected loss functions (i.e., the E[U jx]).12 These functions may be discontinuous and non-monotonic, and can vary in form from obligor i to obligor. More importantly, we have placed no restrictions on the vector of risk factors X. It may be a vectorofanyfinitelengthandwithanydistribution (continuous ordiscrete). Thequantiles ofE[L jX]takeonaparticularly simpleanddesirable asymptotic formwhenweimpose n twoadditional restrictions: (A-3) thesystematicriskfactor X isone-dimensional; and (A-4) thereisanopenintervalBcontaining(cid:11) q (X)andarealnumbern0 <1suchthatforalln > n0,(i) E[L n jx]isnondecreasingonB,and(ii)inf x2B E[L n jx] (cid:21) sup x(cid:20)infB E[L n jx]andsup x2B E[L n jx] (cid:20) inf x(cid:21)supB E[L n jx]. Intuitively, Assumption(A-3)imposesasingleglobalbusinesscycleasthesourceofalldependence across obligors. It is needed in order that (cid:11) (X) be a unique point. Assumption (A-4) is needed so that the q neighborhood oftheq th percentile ofE[L jX]isassociated withtheneighborhood oftheq th percentile of n X. Without (A-4), the tail quantiles of the loss distribution would depend in complex ways both on how conditional expected lossforeachborrowervarieswithx. Amoreparsimonious waytoavoidthisproblem would have been to require that the E[U jx] be nondecreasing in x for all i. However, such a requirement i 11ThedifferencehastodowiththepossibilitythattheunconditionaldistributionsforthefE[L jX]gwillpermitjumppoints(or n arbitrarilysteepslope)atthequantiles(cid:11) (E[L jX])asn ! 1. Thispossibilityispurelyatheoreticalmatter,andwouldnever q n ariseinpracticalapplications. 12Technically,theCreditRisk+modelallowsU toexceedone,becauseitapproximatestheBernoullidistributionofthedefault i eventasaPoissondistribution. ToaccommodateCreditRisk+,wecouldloosenthisrestrictiontoarequirementthattheU have i boundedvariance.Seethemodifiedversionof(A-1)introducedinSection3. 8

wouldexcludehedginginstruments(suchascreditderivatives)andobligorswithcounter-cyclicalcreditrisk. Assumption (A-4) allows for some U to be negatively associated with X, just so long as, asymptotically i andinaggregate,suchinstrumentsdonotalterthemonotonicdependenceoflossesonthesystematicfactor whenX isneartherelevant“tailevent.” Furthermore, (A-4)allowsE[L jx]tobelocally nonmonotonic in n xwhenxisnotintheneighborhood of(cid:11) (X),andallowsfordiscontinuity atanyx. q Fornotational convenience, definefunctions(cid:22) (x) (cid:17) E[U jx]and i i P n (cid:22) (x)A M (x)(cid:17) E[L jx] = iP=1 i i : (9) n n n A i=1 i Wenowhave Proposition 4 If(A-3)and(A-4)aresatisfied,then(cid:11) q (E[L n jX]) = E[L n j(cid:11) q (X)] = M n ((cid:11) q (X))forn> n0. Proof: Fix n > n0. If X (cid:20) (cid:11) q (X), then M n (X) (cid:20) M n ((cid:11) q (X)), so Pr(M n (X) (cid:20) M n ((cid:11) q (X))) (cid:21) Pr(X (cid:20) (cid:11) (X)) (cid:21) q. If M (X) < M ((cid:11) (X)), then X < (cid:11) (X), so Pr(M (X) < M ((cid:11) (X))) (cid:20) q n n q q n n q Pr(X <(cid:11) (X)) < q. Therefore, q inffy : Pr(M (X) (cid:20) y) (cid:21) qg= M ((cid:11) (X)): n n q QED Takentogether,Propositions1,3and4implyasimpleandpowerfulrulefordeterminingcapitalrequirements. Forasseti,setcapital perdollar bookvalue (inclusive ofexpected loss) toc (cid:17) (cid:22) ((cid:11) (X))+(cid:15),for i i q somearbitrarilysmall(cid:15).13 Observethatthiscapitalchargedependsonlyonthecharacteristics ofinstrument iandthusthisruleisportfolio-invariant. Portfoliolossesexceedcapitalifandonlyif Xn Xn U A > c A : (10) i i i i i=1 i=1 Givenourruleforc andthedefinitionofL , i n 0 1 ! ! Xn Xn Xn −1Xn Pr U A > c A = Pr@ L > A ((cid:22) ((cid:11) (X))+(cid:15))A A i i i i n i i q i i=1 i=1 i=1 i=1 = Pr(L > E[L j(cid:11) (X)]+(cid:15)) ! [0;1−q]: n n q Thus, capital is sufficient, in the limit, so that the probability of portfolio credit losses exceeding portfolio capitalisnogreaterthan1−q,asdesired. If additional regularity conditions are imposed in order toeliminate the possibility of discontinuities at the desired quantiles, the insolvency probability converges to 1 − q exactly for (cid:15) = 0. A simple way to achieve this would be torequire that X becontinuous and that the (cid:22) (x) functions be continuous and with i 13In most practitioner discussions, it isassumed that expected loss ischarged against theloan lossreserve and that “capital” refersonlytotheamountheldagainstunexpectedloss.Inthispaper,“capital”referstothegrossamountsetaside. 9

boundedderivatives. However,wecanberatherlessrestrictive, aswereallyneedonlytoguaranteethatthe th asymptoticportfoliolosscdfissmoothandhasboundedderivativesintheneighborhoodofitsq percentile value. Thefollowingcondition issufficienttocircumvent thetechnical caveatsofProposition 3. (A-5) There exists an open interval B containing (cid:11) (X) on which (i) the cdf of systematic factor X is q continuous and increasing, and (ii) for all i, (cid:22) (x) is continuous and differentiable on B, and (iii) i therearerealnumbers(cid:14);(cid:14) andn0 < 1suchthat0 < (cid:14) (cid:20) M n 0(x) (cid:20) (cid:14) < 1foralln > n0. Thisassumption allowsforanon-trivial share oftheportfolio toconsist ofhedging instruments orloans to counter-cyclical borrowers. InAppendixC,Ishowthat Proposition 5 Ifassumptions(A-1)–(A-5)hold,thenPr(L (cid:20) E[L j(cid:11) (X)]) ! qandj(cid:11) (L )−E[L j(cid:11) (X)]j ! 0. n n q q n n q Therefore, for an infinitely fine-grained portfolio, the proposed portfolio-invariant capital rule provides a solvencyprobability ofexactlyq. The results of this section closely parallel recent developments in techniques for capital allocation in a market risk setting. Gourie´roux, Laurent and Scaillet (2000), Tasche (2000) and others show how to take partialfirstderivatives ofVaR.14 Intermsofthenotation usedhere,thefirstderivativeisgivenby d(cid:11) (L ) q n = E[U jL = (cid:11) (L )]: (11) i n q n dA i Under the assumptions of Proposition 5, the condition L = (cid:11) (L ) is asymptotically equivalent to X = n q n (cid:11) (X), which implies that marginal VaR is equal to (cid:22) ((cid:11) (X)). Gourie´roux et al. (2000) require that q i q the joint distribution of the losses fU g be continuous, as otherwise VaR need not be differentiable. This i presents a problem in application to credit risk modeling, as credit risk is largely driven by discrete events (e.g.,defaults). Theapproach takenhereinobtaining Proposition 5allowsfordiscrete (ormixed)U .15 i Portfolio-invariance depends strongly on the asymptotic assumption and on the assumption of a single systematic risk-factor. Portfolios that are not asymptotically fine-grained contain undiversified idiosyncratic risk, which implies that marginal contributions to VaR depend on what else is in the portfolio. As a practical matter, residual idiosyncratic risk is not an impediment to ratings-based capital allocation. Large internationally-active banks aretypically near theasymptotic ideal. Furthermore, thetechniques ofSection 4allowforasimpleportfolio-level correction. Assumption (A-3) is much less innocuous from an empirical point-of-view. It can be relaxed only slightly. Say that some group of obligors shared dependence on a “local” risk-factor. Conditional on X, the fU g within the group would no longer be independent, though they would remain independent of the i fU g outside the group. So long as the within-group exposures in aggregate account for a trivial share of j 14Thisproblemwassolvedindependentlybyseveralauthors.SeereferencesinTasche(2002,x4). 15Tasche(2000)providesslightlylessstringentconditionsfordifferentiability.Tasche(2001)appliesequation(11)toadiscrete model(CreditRisk+),anddiscussesthetechnicalissuesthatarise. 10

thetotalportfolio(i.e.,theycouldbeaggregatedintoasingleexposurewithoutviolatingassumption(A-2)), thelocaldependence canbeignored. Even the largest banks have geographic and industrial concentrations at some level. If these largerscale sectors are not perfectly comonotonic, then portfolio-invariance is lost. Say we had two risk factors, and obligors could differ in their sensitivity to each factor. The realizations (x1;x2 ) associated with a givenquantileofthelossdistribution wouldthendependontheparticularsetofobligorsintheportfolio. In intuitiveterms,theappropriatecapitalchargeforaloantoaheavily-X1-sensitiveborrowerwoulddependon whethertheotherobligorsintheportfoliowerepredominantlysensitivetoX1(inwhichcasetheloanwould addlittlediversificationbenefit)ortoX2 (inwhichcasethediversificationbenefitwouldbelarger). Totake a simple example, let X1 represent the US business cycle and X2 the European business cycle. Consider themergerofastrictlydomesticUS,asymptoticallyfine-grainedportfoliowithanotherasymptoticallyfinegrained bank portfolio. If the second portfolio were also exclusively US, then no diversification benefit would ensue, and required capital for the merged portfolio should be the sum of the capital charges on the two portfolios. However, if the second portfolio contained European obligors, then there would be a diversificationbenefit(aslongasX1 andX2 werenotperfectlycomonotonic), andthemergershouldresult inreduced totalVaR.Therefore, capitalcharges couldnotbeportfolio-invariant. Finally, observe that “bucketing” has not appeared, per se, in the derivation. Indeed, the (cid:22) functions i neednotevenshareacommonformacrossobligors. Sortingobligorsintoafinitenumberofstatisticallyhomogeneousbucketsishelpfulforpurposesofcalibration fromdata,butisnotneededforportfolio-invariant capitalcharges tobeobtained.16 3 Asymptotic loss distribution under mark-to-market valuation Actuarial models are simple to calibrate and understand, and fit naturally with traditional book-value accounting applied tobank loan books. However, muchofthecredit riskismissed, especially forlong-dated highly-rated instruments. Because losses are deemed to arise only in the event of default, no credit loss is recognized when, say, atwo-year AA-ratedloan downgrades after oneyear tograde BB.Underamark-tomarket(MTM)notionofloss,creditriskincludes theriskofdownward(orupward)rating migration, short ofdefault, whentheinstrument’s maturityextendsbeyondtheriskhorizon. Evenforinstitutions thatreport onabook-value basis,itmaybedesirable tocalculate capitalchargeswithinaMTMframeworkinorderto capturetheadditional riskassociated withlongerinstrument maturity. “Loss” is an ambiguous construct in a mark-to-market setting. I follow one widely-used convention in defining the loss rate U on asset i as the difference between expected and realized value at the horizon, i discounted by the risk-free rate and divided by current market value.17 For example, u = 0:2 represents i 16Multi-statemodelssuchasCreditMetricsandCreditPortfolioViewtypicallycalibratePDstoafinitesetofratinggrades,but thefactorloadingsw maybesetattheindividualobligorlevel.Inthiscase,eachobligorwouldcompriseitsown“bucket.”Inthe i KMVmodel,thereisacontinuumof“ratinggrades,”sobucketsdonotariseinanynaturalway. 17Couponpayments,ifany,areassumedtobeaccruedtothehorizonattherisk-freerate.Someconventionalsomustbeimposed onwhichintra-horizoncashflowsarereceivedondefaultingassets. Inpractice,how couponsarehandledhaslittleeffectonthe lossdistribution,andnoqualitativeeffectontheasymptotics. 11

a 20% loss, and u = −0:05 represents a 5% gain. Other definitions can be applied without changing the i resultsbelow. Iredefine“exposure” A asthecurrentmarketvalue. i Credit risk arises due to uncertainty in U. As before, I assume a vector of systematic risk factors X andthattheU areconditionally independent. Theparameterization andcalibration ofthe(cid:22) (x) (cid:17) E[U jx] i i i functions can draw on existing industry models such as CreditMetrics. Say, for example, that we have a rating system with G non-default grades (grade G+1 denoting default), and for each obligor i we have a setofunconditional transitionprobabilitiesp(cid:22) forgradegatthehorizon. Fromthesewecalculatethreshold ig values γ for obligor i’s asset return R (see equation (2)), such that obligor i defaults if R (cid:20) γ , and ig i i i;G transitsto“live”grade g ifγ i;g < R i (cid:20) γ i;g−1. Thevariables(X;(cid:15)1;(cid:15)2;::: ;(cid:15) n )areiidN(0;1). Therefore, theconditional transition probabilities aregiveninCreditMetrics by (cid:18) q (cid:19) (cid:18) q (cid:19) p ig (x) = (cid:8) (γ i;g−1 +xw i )= 1−w i 2 −(cid:8) (γ i;g +xw i )= 1−w i 2 ; (12) andtheunconditionaltransitionprobabilitiesdeterminethethresholdsas γ i;g = (cid:8)−1(p(cid:22) i;g+1 +:::+p(cid:22) i;G+1 ). Consider a zero-coupon instrument maturing at or after the horizon. Assume the current value A is i known, and let v (x) be the value of instrument i at the horizon conditional on the obligor migrating to i;g ratingg. Instandard implementations ofCreditMetrics, pricingatthehorizonisdonebydiscounting future contractualcashflows,wherethespreadsforeachgradearetakenasfixedandknown. Inprinciple,however, wecan allow spreads to benon-stochastic functions ofX. Theconditional expected mark-to-market value atthehorizon is XG MTM i (x) = v ig (x)p ig (x)+A (cid:22) i (1−E[LGD i jx])p i;G+1 (x); (13) g=1 (cid:22) whereA isthesizeofthebank’slegalclaimontheobligorintheeventofadefault. Couponscaneasilybe i accommodated inthispricingformulaaswellwithsomeadditional notation. Theconditional expected loss functions (cid:22) (x)arethengivenby i exp(−rT ) (cid:22) (x) = h (E[MTM (X)]−MTM (x)); (14) i i i A i whereT isthetimetohorizon andristherisk-free yieldforterm T . h h The results of the previous section can be adapted to a mark-to-market setting without difficulty. In contrast to the actuarial case, MTM loss is not bounded from below by zero (e.g., if the obligor’s rating improves,theretypically willbeagaininvalue). Inprinciple, itneednotbeboundedfromaboveeither. To accommodate theMTMcase,Imodifyassumption (A-1)asfollows: (A-1) Conditional on X, the fU g are independent. The conditional second moment of loss exists and is i bounded; i.e., there exists afunction (cid:7)(x)such that E[U 2jx] (cid:20) (cid:7)(x) < 1for all instruments iand i realizations x. Furthermore, E[(cid:7)(X)] < 1. Thisversionoftheassumption isstrictlyweakerthantheversionofSection1. 12

Foragivenportfolioofnassets,L ,asdefinedinequation(4),isthediscountedportfoliomarket-valued n creditlossatthehorizonasapercentageofcurrentmarketvalue. IfindthatallofthePropositionsofSection 2 continue to hold, as stated, under the relaxed version of assumption (A-1). Indeed, the proofs given in the appendix explicitly rely only on the relaxed version. The results in no way depend on the assumptions andconventions ofCreditMetrics, whicharedescribed aboveforillustrative purposes.18 Bythesamelogic as before, the appropriate asymptotic capital charge per dollar current market value for asset i is simply (cid:22) ((cid:11) (X)). i q 4 Capital adjustments for undiversified idiosyncratic risk Noportfolio iseverinfinitelyfine-grained: real-worldportfolios havefinitenumbersofobligors andlumpy distributions of exposure sizes. Large portfolios of consumer loans ought to come close enough to the asymptotic ideal that this issue can safely be ignored, but we ought not to presume the same for even the largest commercial loan portfolios. Unless ratings-based capital rules are tobe abandoned forafull-blown internal models approach, we require a methodology for assessing a capital add-on to cover the residual idiosyncratic riskthatremainsundiversified inaportfolio. Consider a homogeneous portfolio in which each instrument has the same conditional expected loss function (cid:22)(x) and the same exposure size. Under assumptions (A-3) and (A-4) and suitable regularity conditions, (cid:11) (L )= (cid:22)((cid:11) (X))+O(n −1): (15) q n q Thatis,thedifferencebetweentheVaRforagivenfinitehomogeneous portfolioanditsasymptoticapproximationisproportional to 1=n. One way to obtain this result is through a generalized Cornish-Fisher expansion due to Hill and Davis (1968) for a sequence of distributions converging to an arbitrarily differentiable limiting distribution. The j th termintheexpansionof(cid:11) (L )isproportionaltothedifferencebetweenthej th cumulantsofthedistriq n butionsforL n andL1. Underverygeneralconditions, thecumulants(forj (cid:21) 2)converge atO(n −1). The difficulty isinspecifying precisely asetofregularity conditions underwhichtheCornish-Fisher expansion isguaranteed tobeconvergent. BuildingontheresultsofGourie´rouxetal.(2000),MartinandWilde(forthcoming)deriveequation(15) more rigorously as a Taylor series expansion of VaR around its asymptotic value. Although the necessary regularityconditionsremainslightlyopaque,themainadditionalrequirementisthattheconditionalvariance V[Ujx]islocally continuous anddifferentiable inx. Furthermore, MartinandWildeshowthattheO(n −1) 18InthespiritofKMVPortfolioManager, forexample, onecouldreplaceequation(12)withtheconditional densityfunction forthedefaultprobabilityatthehorizon. Thesummationinequation(13)wouldbereplacedbyanintegral,andthev wouldbe ig obtainedusingrisk-neutralvaluation.Valuationinthedefaultstateinequation(13)alsowouldbemodified. 13

termisgivenby(cid:12)=nwhere19 (cid:12) (cid:18) (cid:19)(cid:12) −1 d V[Ujx]h(x) (cid:12) (cid:12) = (cid:12) (16) 2h(x)dx (cid:22)0(x) (cid:12) x=(cid:11)q (X) andwhereh(x)isthepdfofX. Of course, equation (15) is itself an asymptotic result. When we say that convergence is at rate 1=n, we are saying that for large enough n the gap between VaR and its asymptotic approximation shrinks by halfwhennisdoubled. Shortofrunning thecredit VaRmodel, thereisnowaytosaywhetheragivennis “large enough” for this relationship to hold. Tosee whether our “1=n rule” works well for realistic values + ofnandrealisticmodelcalibrations, Iexaminethebehavior ofVaRinanextendedversionofCreditRisk . + The virtue of CreditRisk for this exercise is that it has an analytic solution. We not only can execute the model for any n very quickly, but also avoid Monte Carlo simulation noise in the results. However, the + standardCreditRisk modelassumesfixedlossgivendefault,andsoignores apotentially importantsource of volatility.20 For the special case of a homogeneous portfolio, it is not difficult to augment the model to allowforidiosyncratic recoveryrisk. + As in the standard CreditRisk , assume that the systematic risk factor X is gamma-distributed with mean one and variance (cid:27) 2 . Each obligor has the same default probability p(cid:22)and factor loading w. Each facilityintheportfoliohasidenticalexposuresize,whichisnormalizedtoone,andidenticalexpectedLGD. Thefunctional formforconditional expected lossfunction is (cid:22)(x) = E[LGD](cid:1)p(cid:22)(1+w(x−1)): (17) Tointroduce idiosyncratic recoveryrisk,assumeLGDforeachobligorisdrawnfromagammadistribution 2 with mean (cid:21) and variance (cid:17) . This specification is convenient because the sum of m independent and 2 identical gammarandom variables isgamma-distributed withmean m(cid:21)and variance m(cid:17) . LetG denote m the gammacdf withthis mean and variance. Let(cid:25) denote theprobability that there willbe m defaults in m + theportfolio; these probabilities arecalculated intheusual wayinCreditRisk . ThecdfofL can thenbe n obtained as X1 Pr(L (cid:20) y)= (cid:25) G (ny): (18) n m m m=0 Long before m approaches n, the (cid:25) become negligibly small, so numerical calculation of equation (18) m presents no difficulty. A minor disadvantage of this specification is that it allows LGD to exceed one. However,solongas(cid:17)isnottoolarge,aggregatelossesintheportfoliowillbewell-behaved,sotheproblem canbeignored. 19Equation(16)isobtainedthroughlessformalargumentsinWilde(2001). 20Thestandardmodelalsoimpliesadiscretelossdistribution. Asnincreases,the“steps”inthelossdistributionarere-aligned, whichcauseslocalviolationsofmonotonicityintherelationshipbetweennandVaR. 14

Forthismodel,theasymptotic slope(cid:12) isgivenby (cid:18) (cid:18) (cid:19)(cid:18) (cid:19) (cid:19) 1 1 (cid:27) 2−1 1−w (cid:12) = ((cid:21) 2+(cid:17) 2) 1+ (cid:11) (X)+ −1 (19) 2(cid:21) (cid:27)2 (cid:11) (X) q w q Thisformulageneralizes aformula derivedinWilde(2001) under thespecificparameter values usedinthe Baselproposal. + Calibration is intended to be qualitatively faithful to available data. When CreditRisk is calibrated to rating agency historical performance data, as in Gordy (2000), one finds a negative relationship between p(cid:22) andw. Bycontrast,whenaMertonmodelsuchasCreditMetricsiscalibratedtothesedata,thereisnostrong relationship between PD and factor loading. This makes sense, as there is no strong reason to expect that averageasset-valuecorrelationshouldvarysystematicallyacrossratinggrades. Tomakeuseofthisstylized fact in our calibration, I choose a constant asset-value correlation of 15% in CreditMetrics, and calculate + a within-grade default correlation for each grade. Shifting back to CreditRisk , I set a conservative but reasonable value of (cid:27) = 2 for the volatility of X, and then calibrate w for each rating grade so that the within-grade default correlation matches the value from CreditMetrics.21 The remainder of the calibration exercise isstraightforward. Ichoose stylized values for thedefault probabilities, andassume thatLGDhas mean0.5andstandard deviation 0.25. Thechosencoveragetargetisq = 0:995ofthelossdistribution. Results areshowninTable1forfiverating grades. Thefinalcolumn(n = 1)provides theasymptotic capitalcharge,sothedifferencebetweeneachcolumnandthefinalcolumnrepresentsthe“true”granularity add-on. Even for portfolios of only n = 200 homogeneous obligors, granularity add-ons are small in the absolute sense(under 60basispoints). However,theadd-ons canbelargerelative totheasymptotic capital charge for investment grade obligors. For a homogeneous portfolio of 200 A-rated loans, the granularity add-onisroughlyequaltotheasymptotic charge. (cid:3) Table1: Convergence ofValue-at-Risk VaR [L ]forvaluesofn q n p(cid:22) w 200 500 1000 2000 5000 1 A 0:06 1:011 0:723 0:521 0:445 0:406 0:381 0:364 BBB 0:20 0:836 1:425 1:190 1:106 1:064 1:038 1:020 BB 1:25 0:602 5:217 4:947 4:856 4:810 4:783 4:764 B 6:25 0:415 17:881 17:584 17:485 17:435 17:405 17:385 CCC 17:50 0:295 37:663 37:335 37:226 37:172 37:139 37:117 *:DefaultprobabilitiesandVaRexpressedinpercentagepoints.Simulationsassumeq=0:995,(cid:27)=2, (cid:21)=0:5and(cid:17)=0:25. Figure1demonstrates therelationship betweenthetheoretical granularity add-on and1=nforthreehomogeneousportfolios. Foranextremelylowqualityportfolio(CCCrating),thepredictedlinearrelationship 21SeeGordy(2000)formoredetailsonthechoiceof(cid:27)andonusingwithin-gradedefaultcorrelationsforconsistentcalibration acrossthetwomodels. 15

holds down to n = 200.22 For the medium quality (BB rated) portfolio, there are visible but negligible departures from the predicted linear relationship when n < 500. For a high quality portfolio (A rated), departures from linearity are visible at n = 1000 and become significant at lower values of n. Because departures from linearity areintheconcave direction, agranularity adjustment calibrated totheasymptotic slopewouldslightlyovershoot thetheoretically optimaladd-on forsmallerhigh-quality portfolios. Figure1: Granularity Add-onasLinearFunctionof1=n 0.5 0.4 0.3 0.2 0.1 0 1/5000 1/2000 1/1000 1/500 1/300 1/200 1/n no−ddA ytiralunarG o CCC x BB + A Note:The“true” granularity add-on in theextended CreditRisk+ model isplotted withsymbols for threehomogeneous portfoliosofvarioussizes.Thelinesshowthecorrespondingtheoreticaladd-onforthismodel. In the case of a non-homogeneous portfolio, determining an appropriate granularity add-on is only slightly more complex. The method of Wilde (2001) accommodates heterogeneity (the V[Ujx] and (cid:22)(x) terms in equation (16) become V[L jx] and M (x), respectively). An alternative two-step method also n n appears to work quite well and may be better suited to a regulatory setting. The first step is to map the actual portfolio to a homogeneous “comparable portfolio” by matching moments of the loss distribution. 22TheslopebetweeneachplottedpointisconstanttosixsignificantdigitsforbothB(notshown)andCCCportfolios. 16

The second step is to determine the granularity add-on for the comparable portfolio. The same add-on is appliedtothecapitalchargefortheactualportfolio. ConsideraheterogeneousportfolioofnlendingfacilitiesdividedamongBbuckets. Withineachbucket b, every facility has thesame PDp(cid:22) ,factor loading w , expected LGD(cid:21) and LGDvolatility (cid:17) . Exposure b b b b sizes A areallowedtovaryacross facilities inabucket. Tomeasure theextenttowhichbucket bexposure i isconcentrated inasmallnumberoffacilities, werequirethewithin-bucket Herfindahlindexgivenby23 P 2 A H (cid:17) (cid:0)Pi2b i(cid:1) : b 2 A i2b i ThehigherisH ,themoreconcentrated istheexposurewithinthebucket, sothemoreslowlyidiosyncratic b risk isdiversified away. Thematching methodology takes bucket-level inputs fp(cid:22) ;w ;(cid:21) ;(cid:17) ;H g andtotal b b b b b bucket exposure. Thisdatastructure maybeespecially convenient inaregulatory setting withbucket-level reporting requirements. (cid:3) Thegoalistoconstructthecomparableportfolioasaportfolioofn equal-sizedfacilitieswithcommon PDp(cid:22)(cid:3) ,factorloadingw (cid:3) ,andLGDparameters(cid:21) (cid:3) and(cid:17) (cid:3) . Inprinciple,awidevarietyofmomentrestrictions couldbeusedtodothemapping,butitseemsbesttochoosemomentswithintuitiveinterpretation. Appendix Ddevelops amatchingprocedure basedonfivemoments:24 (cid:15) exposure-weighted expected defaultrate, (cid:15) expected portfolio lossrate, (cid:15) contribution ofsystematic risktolossvariance, (cid:15) contribution ofidiosyncratic default risktolossvariance, and (cid:15) contribution ofidiosyncratic recoveryrisktolossvariance. Under this methodology, each of the parameters of the comparable portfolio is given by an explicit linear equation thatcanbeinterpreted asaweighted average ofthecharacteristics oftheheterogeneous portfolio. (cid:3) Most interestingly, the number of loans n can be interpreted as an inverse measure of weighted exposure (cid:3) concentration. Finally, the asymptotic slope (cid:12) for the comparable homogeneous portfolio is given by equation (19). The portfolio data needed for the mapping method should pose minimal additional reporting burden for regulated institutions. Default probability, expected LGD and total bucket exposure would need to be reported by the bank to calculate the asymptotic capital charge. Factor loadings and LGD volatilities would likely be assigned as functions of the p(cid:22) and (cid:21) ; this is indeed the case in Basel Committee on b b Bank Supervision (2001). The only new required inputs, the within-bucket Herfindahl indices, are easily calculated fromtheindividual exposuresizes. 23The Herfindahl index isameasure of concentration inverywidespread use inanti-trust analysis, andshould befamiliar to manypractitioners. 24The matching procedure specified in the proposed granularity adjustment of Basel Committee on Bank Supervision (2001, Chapter8)isbasedonadifferentsetofmomentsbutfollowssimilarintuition. 17

Matchinglower-ordermomentsgivesnoguaranteethatthelossdistributionforthecomparableportfolio willdisplayhigher-ordermomentsveryclosetothoseoftheoriginalheterogeneousportfolio. Tailquantiles ofthelossdistribution aresensitivetohigher-order moments,sotheperformanceofthemethodologyneeds tobeconfirmedonarangeofempiricallyplausibleportfolios. Asanexample,Iconstructaportfolioof600 obligors divided equally across four buckets. Thebuckets represent high investment grade, low investment grade, high speculative grade and moderate-to-low speculative grade. Factor loadings are calibrated as in Table1. ExpectedLGDsforthebucketsaresetto0.3,0.2,0.6,and0.5,respectively, andtheLGDvolatility p issetto(cid:17) = 0:5 (cid:21) (1−(cid:21) ). Table2displays thebucket-level parameters. b b b (cid:3) Table2: Bucket-levelParametersofStylizedPortfolio p(cid:22) w (cid:21) (cid:17) 1 0:05 1:040 0:3 0:229 2 0:50 0:715 0:2 0:200 3 1:00 0:629 0:6 0:245 4 5:00 0:440 0:5 0:250 *:Defaultprobabilitiesinpercentagepoints. 4 Exposure size for facility i is set to i ; i.e., A is $1 for the first exposure, $16 for the second, $81 for i thethird,andsoon. Theexposuresareassignedtobucketsbyturn. ThefirstexposureisassignedtoBucket 4, the second to Bucket 3, the third to Bucket 2, the fourth to Bucket 1, the fifth to Bucket 4, and so on. Looking at the portfolio as a whole, I find that the largest 10% of exposures account for roughly 40% of total exposure, which matches the empirical rule of thumb reported by Carey (2001) for concentration of outstandings. Also, portfolio exposure is roughly split between investment and speculative grades, which appearstobetypical ofacommercialloanportfolio atalargebank.25 I first obtain parameters for the comparable homogeneous portfolio. The comparable portfolio has n (cid:3) = 218:7 obligors, which is under 40% of the obligor count of the original portfolio.26 Each obligor hasPDof1.64%andfactorloadingw (cid:3) = 0:487. Lossgivendefaulthasexpectedvalue0.491andvolatility 0.247.27 By construction, the comparable portfolio matches the original portfolio in its expected loss rate of 0.804%. For each portfolio, the standard deviation of the loss rate is 0.918%. Once the comparable portfolio is determined, the constant of proportionality (cid:12) is calculated for the target percentile q. Finally, I approximate VaR fortheoriginal portfolio asitsasymptotic VaRplus(cid:12) (cid:3) =n (cid:3) . q q Results are shown in Table 3 for three tail values of q. Row (i) presents estimates of VaR obtained by direct simulation of the original portfolio. Row (ii) presents the asymptotic VaR for the original portfolio givenbyE[L j(cid:11) (X)]. Row(iii)showsVaRforthecomparableportfolioobtainedusingthecdfinequation n q 25Inasampleoflargebankcommercialloanportfolios,TreacyandCarey(1998,Chart3)showthatroughlyhalfofaggregate internallyratedoutstandingsareinvestmentgrade. 26Notethattheprocedureforcalculatingthegranularityadd-ondoesnotrequiren(cid:3)tobeaninteger. 27Inpractice,asinthisexercise,LGDvolatilityisoftenassumedtobeasimplefunctionofexpectedLGD.Reportingrequirementsandcomputationscouldbesimplifiedbysetting(cid:17)(cid:3) =VLGD((cid:21)(cid:3)). WhilethisignorestheeffectofnonlinearityinVLGD, thedifferenceistypicallysmall.Inourexample,VLGD((cid:21)(cid:3))wouldhavebeen0.250. 18

(cid:3) (cid:3) (18). Thegranularity add-on(cid:12) =n isshowninrow(iv). Row(v)sumstheasymptoticVaRandgranularity q add-ontogetourapproximation. Trackingerrorbetweenrows(v)and(i)isshowninthefinalrow. The procedure works well for all values of q. Despite the relatively small obligor count in the comparable portfolio, the error due to linear approximation of the “1=n rule” is minimal. At q = 99:5%, our approximated VaRovershoots itstarget by2.2basispoints. (cid:3) Table3: DirectandApproximated EstimatesofVaR q: 99.0 99.5 99.9 (i) “True”VaR 4:577 5:522 7:872 (ii) AsymptoticVaR 4:220 5:109 7:260 (iii) VaRforcomparable portfolio 4:570 5:535 7:872 (iv) Granularity add-on 0:357 0:435 0:627 (v) Approximated VaR 4:578 5:544 7:886 (vi) Trackingerror 0:001 0:022 0:014 *:Allquantitiesexpressedinpercentagepoints.“True”VaRestimatedbysimulation with300,000MonteCarlotrials. Itshould beemphasized that thetheoretical underpinnings forthegranularity adjustment applyequally tomark-to-marketmodels. Thesimplelinearformulaeforparametersofthecomparablehomogeneousport- + foliodependonthelinearfunctionalformsassumedinCreditRisk . Specificationsbasedonmorecomplex models, e.g.,KMVPortfolio Manager orCreditMetrics, implymorecomplexmapping formulae whoseinputs need not bereducable tobucket-level summary statistics (e.g., Herfindahl indices). However, itseems + reasonabletoconjecturethatonecanachievetolerableaccuracyusingcruderulesbasedontheCreditRisk formulae. Whatismostimportant isthattherebeareasonably accurate measure forthe“effective” obligor (cid:3) count (i.e., n ) in a heterogeneous portfolio. Most bank portfolios are heavy-tailed in exposure size distri- (cid:3) bution, and thus may have an effective n that is an order of magnitude smaller than the raw obligor count intheportfolio. 5 Asymptotic properties of alternative risk measures Industry application ofcreditriskmodelingtocapitalallocation appears almostinvariably toequate soundness withacoverage target for value-at-risk. However, because it ignores the distribution of losses beyond the target quantile, VaR has significant theoretical and practical shortcomings. As has been emphasized in recentliteratureonrisk-measures, VaRisnotsub-additive. Thatis,ifL andL arelossesonbankportfo- A B liosAandB,thenweneednothaveVaR [L +L ] (cid:20) VaR [L ]+VaR [L ],whichwouldimplythata q A B q A q B merger ofbank Aand bankBcouldincrease VaR;seeFreyandMcNeil(2002, x2.3)foranexample based oncreditriskmeasurement. Sub-additivity isoneofthefourrequirementsfora“coherent” riskmeasure,as definedbyArtzner,Delbaen, EberandHeath(1999). Undertheassumptionsneededtoachieveportfolioinvariance,wehaveVaR [L +L ]= VaR [L ]+ q A B q A VaR [L ],sosub-additivity ispreserved inmergers ofasymptotically fine-grained portfolios. Eveninthis q B 19

case, however, VaR can be manipulated by splitting the portfolio. Under an actuarial definition of loss, for example, aportfolio consisting ofasingle loanwithdefault probability under1−q hasVaR = 0, butthe q sameloanhaspositivecontribution (assumingpositivedependence ofU onX)toVaRinanasymptotically fine-grainedportfolio. SegregatingthisloanfromthelargerportfoliodoesnotchangetheVaRcontributions oftheremaining loans,soVaRisunambiguously reduced. Anotherproblemisthatamean-preservingspreadofthelossdistributioncandecreaseVaR.Thisresults inacounterintuitivenon-monotonicrelationshipbetweenwithin-portfoliocorrelationandVaR.Correlations increase with factor loadings, and, when factor loadings are low to moderate, VaR does as well. However, asfactorloadingsarepushedhigherandhigher, thelossdistribution becomesincreasingly long-tailed. VaR thenshrinkstowardsthemedian,whiletheprobabilityofacataclysmiclossincreases. Inthelimitingcaseof perfectlycomonotoniclosses,thedefaultrateiseitherzero(withprobability 1−p(cid:22))orone(withprobability p(cid:22)). Ifp(cid:22)< 1−q,thenVaR = 0. Forasurveydiscussion ofthepotential pitfallsofVaR,seeSzego¨ (2002). q AsanalternativetoVaR,AcerbiandTasche(2002)proposeusinggeneralizedexpectedshortfall(“ES”), definedby (cid:0) (cid:1) ES q [Y]= (1−q)−1 E[Y (cid:1)1 fY(cid:21)(cid:11)q (Y)g ]+(cid:11) q (Y)(q−Pr(Y < (cid:11) q (Y))) : (20) Thefirsttermisoftenusedasthedefinitionofexpectedshortfallforcontinuousvariables. Itisalsoknownas th “tailconditionalexpectations.” Thesecondtermisacorrectionformassatthe q quantileofY. Underthis definition, Acerbi andTasche(2002) show thatESiscoherent andequivalent toRockafellar andUryasev’s (2002)CVaR. Expected shortfall offers some important advantages as a soundness standard. By Acerbi and Tasche (2002, Corollary 3.3), ES is continuous and monotonic is q, so small increases in the “stringency” of the q capital rule (as controlled by q) lead to small increases in required capital. ES is nondecreasing with any mean-preserving spreadinthelossdistribution, soESincreases withfactorloadings. InAppendixE,Ishowthat Proposition 6 Ifassumptions(A-1)-(A-5)hold,thenjES [L ]−ES [M (X)]j ! 0. q n q n An immediate implication is that ES-based capital charges are portfolio invariant under the same assumptions as VaR-based capital charges. The asymptotic expected shortfall, ES [M (X)], can be decomposed P P q n as( c A )=( A ),wherethecapitalcharge perdollarofexposure toiisc = E[U jX (cid:21) (cid:11) (X)]. Obi i i i i i i q servethatc depends onlyonhowU depends onX,andsoisportfolio-invariant. Underavarietyofmodel i i specifications, c hasan analytical solution, sothere isnooperational difficulty incalibrating ratings-based i capitalcharges toanESsoundness standard. AnotheralternativetoVaRisexpectedexcessloss(“EEL”).ForarandomvariableY andtargetlossrate (cid:18) > 0,EELisdefinedby EEL [Y] (cid:17) inffy :E[(Y −y)+] (cid:20) (cid:18)g: (21) (cid:18) 20

whereY + denotes max(Y;0). Underthe EELparadigm, aninstitution holds capital (plus reserves) sothat theexpected credit loss inexcess ofcapital isless thanorequal tothetarget loss rate. Thatis, therequired totalcapital(plusreserves)isgivenby c= EEL [L ]perdollaroftotalexposure. (cid:18) n EEL is sensitive to the tail of the loss distribution, so shares many of the advantages of ES. More importantly, the target rate (cid:18) represents the expected loss borne by the depository insurance agency (such astheFDICintheUS),soEEL-basedcapital hasanatural policy interpretation. UnlikeES,however, EEL cannotbereconciled withportfolio-invariant capitalcharges. Someintuitionforthisproblemcanbegained bywritingtheasymptoticEELforhomogeneousportfoliosintermsofthedistributionofthesystematicrisk factor. Assume we have loans of two types, denoted “a” and “b”. Let (cid:22) (x) denote the expected loss for a bucketaloansconditional onX=x. Byreasoning verysimilartothatofProposition 6,wecanshowthat E[(L −y)+] ! E[((cid:22) (X)−y)+] a a foranyy. Therefore,theasymptoticEELcapitalchargec issetsothatE[((cid:22) (X)−c )+]equalsthedesired a a a target(cid:18). Similaranalysisforbucketbgivesc . b Now say we have a mixed portfolio containing equal numbers of loans from a and b. For simplicity, the exposures are equal-sized. Asymptotic EEL capital for the mixed portfolio is given by EEL [(cid:22) (X)]. (cid:18) m By construction of the mixed portfolio, we have (cid:22) (X) = ((cid:22) (X)+(cid:22) (X))=2. If asymptotic EELwere m a b portfolio-invariant, thenc (cid:17) (c +c )=2wouldsatisfy m a b (cid:18) = E[((cid:22) (X)−c )+]: (22) m m Wenowrequirethefollowingtriangleinequality: Lemma1 IfY1andY2areintegrablerandomvariablesonaprobabilityspace(Ω;F;P),then E[(Y1 +Y2 )+] (cid:20) E[Y 1 +]+E[Y 2 +]: (23) IfP(f! : (Y1 (!)<0<Y2 (!))_(Y2 (!)<0<Y1 (!))g) > 0,thentheinequalityinequation(23)isstrict. ProofisgiveninAppendixF. Theconditions ofLemma1applytoY (cid:17) ((cid:22) (X)−c )=2,whichgivesus j j j E[((cid:22) (X)−c )+] (cid:20) E[(((cid:22) (X)−c )=2)+]+E[(((cid:22) (X)−c )=2)+] =(cid:18): (24) m m a a b b In general, the threshold realization of X at which (cid:22) (x) = c does not equal the corresponding threshold a a for portfolio b, so for some interval of x values we will have either (cid:22) (x) − c < 0 < (cid:22) (x) − c or a a b b (cid:22) (x)−c > 0 > (cid:22) (x)−c . Therefore, the inequality in equation (24) will in most situations be strict, a a b b whichimpliesthatc istoostrictacapitalrequirement fortheasymptoticmixedportfolio. m Toprovidearoughideaofhowmuchweovershootrequiredcapitalinamixedportfolio, IapplyEELto + anasymptotic,singlesystematicfactorversionofCreditRisk . InAppendixG,IshowthatasymptoticEEL 21

takes onarelatively simple forminthismodel. Table4presents EEL-andVaR-based capital requirements for homogeneous asymptotic portfolios of different credit ratings. Parameters for each rating grade and the volatility of X are taken from Table 1. The “EEL” and “VaR” columns in Table 4 report required capital charges (gross of reserves) for an EEL target of (cid:18) = 0:00002 (i.e., 0.2 basis points) and a VaR target of q = 99:5%, respectively. Thevalue of (cid:18) waschosen to equate capital requirements under the two standards for an obligor at the border of investment and speculative grades (i.e., between BBB and BB). In this example, the EEL standard produces lower (higher) capital requirements than VaR for the higher (lower)grades. (cid:3) Table4: AsymptoticEELandVaRCapitalCharges EEL VaR AA 0:050 0:135 A 0:131 0:248 BBB 0:571 0:709 BB 4:135 3:397 B 19:352 12:657 CCC 45:550 27:390 *:Capitalinpercentagepoints. I next form mixed portfolios. In each case, I assume an asymptotic portfolio of equal-sized loans, half of which are in one bucket and half in another bucket. It is straightforward to show that the conditional expectedlossrateforamixedportfolio is 1 1 (cid:22) (x) = (cid:22) (x)+ (cid:22) (x) = (cid:21)p(cid:22) (1−w +w x) (25) m 2 a 2 b m m m where p(cid:22) = (p(cid:22) + p(cid:22) )=2 and w = (p(cid:22) w + p(cid:22) w )=(2p(cid:22) ). The (cid:22) (x) take on the same form as for m a b m a a b b m m homogeneous portfolios, sothe tools ofAppendix Gapply without modification. Results forfour different mixedportfoliosarepresentedinTable5. ThethirdcolumnshowstheEELforthemixedportfolio,whilethe fourth column shows the average of the EELs for homogeneous portfolios of the two constituent buckets. The final column shows the “tracking error” as a percentage of the third column. As one would expect, the average of the homogeneous capital charges overshoots the correct mixed-portfolio capital charge by a relativelysmall(thoughnon-negligible) amountwhenthetwobucketsareadjacent. ForamixofgradesAA andA,weovershootbyunder3%. ForamixofBBBandBB,weovershootby6.5%. Ifdistantbucketsare mixed,theovershoot ismuchlarger(over16%forthetwoexamplesinthetable). Discussion Thispapershowshowrisk-factormodelsofcreditvalue-at-riskcanbeusedtojustifyandcalibratearatingsbasedsystemforassigning capitalchargesforcreditriskattheinstrumentlevel. Ratings-based systems,by definition,permitcapitalchargestodependonlyonthecharacteristics oftheinstrumentanditsobligor, and 22

Table5: AsymptoticEELforMixedPortfolios Bucketa Bucketb c (c +c )=2 Error m a b AA A 0:088 0:090 +2.7% A B 8:378 9:741 +16.3% BBB BB 2:210 2:353 +6.5% AA CCC 19:658 22:800 +16.0% *:Capitalchargesexpressedinpercentagepoints. not the characteristics of the remainder of the portfolio. Risk-factor models deliver this property, which I call portfolio invariance, only if two conditions are satisfied. First, the portfolio must be asymptotically fine-grained, in order that all idiosyncratic risk be diversified away. Second, there can be only a single systematic riskfactor. Violation ofthe firstcondition, which occurs foreveryfinite portfolio, does not pose aserious obstacle in practice. Analysis of rates of convergence of VaR to its asymptotic limit leads to a robust and practical methodofapproximating aportfolio-level adjustment forundiversified idiosyncratic risk. Thesecond condition presents agreater dilemma. Thesingle risk factor assumption, ineffect, imposes asinglemonolithicbusinesscycleonallobligors. ArevisedBaselAccordmustapplytothelargestinternationalbanks,sothesingleriskfactorshouldinprinciplerepresenttheglobalbusinesscycle. Byassumption, allothercreditriskisstrictlyidiosyncratic totheobligor. Inreality,theglobalbusinesscycleisacomposite ofamultiplicityofcyclestiedtogeographyandtopricesofproductioninputs. Asinglefactormodelcannot capture any clustering of firm defaults due to common sensitivity to these smaller-scale components of the globalbusiness cycle. Holdingfixedthestateoftheglobaleconomy,localeventsin,forexample,Spainare permitted to contribute nothing to the default rate of Spanish obligors. If there are indeed pockets of risk, then calibrating a single factor model to a broadly diversified international credit index may significantly understate thecapitalneededtosupportaregional orspecialized lender. Would empirical violation of the single factor assumption necessarily render a risk-bucket capital rule unreliable and ineffective? The answer depends on the scope of application and the sophistication of debt markets. Regulators will need to use caution and judgement in applying risk-bucket capital charges to institutions that are less broadly diversified. One should note that the current Basel Accord, which is itself a risk-bucket system, is applied to an enormous range of institutions, so it seems unlikely that a reformed Accordwouldbringaboutanygreaterharm. More generally, the ability of banks to subvert ratings-based capital rules by exploiting the inadequacy of the single factor assumption depends on the capacity of debt markets to recognize and price different risk-factors. At present, such capacity appears to be lacking. Partly because markets do not yet provide precise information on correlations of credit events across obligors, many (perhaps most) of the institutions that actively use credit VaR models effectively impose the single-factor assumption.28 In the near- to 28UsersofKMVPortfolioManagerandCreditMetricsoftenimposeauniformasset-valuecorrelationacrossobligors. Usersof CreditRisk+ typicallyassumeasinglefactorandafactorloadingof w = 1forallobligors. Inboththeseexamples, theuseris 23

medium-term, therefore, the implausibility of the single factor assumption need not present an obstacle to the implementation of reformed ratings-based risk-bucket capital rules. In the long run, however, the need torelaxthisassumption mayimpeladoption ofamoresophisticated internal-models regulatory regime. Appendix A Proof of Propositions 1 and 2 TheproofofProposition 1requiresaversionofthethestronglawoflargenumbersforasequence fY gof n independent random variablesandasequence fa gofpositiveconstants: n Lemma2(Petrov(1995), Theorem6.7) P Ifa "1and 1 (V[Y ]=a 2) < 1,then n n=1 n n " #! 1 Xn Xn Y −E Y ! 0a:s:: n n a n i=1 i=1 Wealsomakeuseofthefollowinglemma: Lemma3 P Iffb gisasequenceofpositiverealnumberssuchthatfb gisO(n −(cid:26))forsome(cid:26)> 1,then 1 b < 1. n n n=1 n This lemma is a corollary of Theorem 3.5.2 in Knopp (1956) and the convergence of the harmonic series 1=n(cid:26) for(cid:26)> 1(seeKnopp1956,Example3.1.2.3). P We now prove Proposition 1. Let Y (cid:17) U A and a (cid:17) n A . For any realization x, conditional n n n n i=1 i independence implies ! X1 X1 Xn 2 (V[Y jx]=a 2)= A = A V[U jx] n n n i n n=1 n=1 i=1 Undertheactuarial definition ofloss, U isbounded in[0;1], sowemusthaveV[U jx] < 1foranyX=x. n n Forthispropositiontoholdunderthemark-to-marketparadigmaswell,assumption(A-1)providesabound onV[U jx]. Therefore, undereitherdefinitionofloss,thereexistsafiniteconstant V (cid:3) suchthat n ! X1 X1 Xn 2 (V[Y jx]=a 2)(cid:20) V (cid:3) A = A : n n n i n=1 n=1 i=1 P By part (bn) of ass P umption (A o-2), the sequence fA n = n i=1 A i g is O(n −(1=2+(cid:16))) for some (cid:16) > 0, so the sequence (A = n A )2 is O(n −(1+2(cid:16))). By Lemma 3, the series sum must be finite. By part (a) of n i=1 i implicitlyimposingbothasinglesystematicfactorandauniformvalueforthefactorloading. 24

assumption (A-2), we have a " 1. The conditions of Lemma 2 are thus satisfied. The loss ratio L is P n n equalto n Y =a ,soProposition 1isproved. QED i=1 i n Proposition 2followssimilarlogic. Werequirethefollowinglemma: Lemma4 P Letfb g andfd g besequencesofrealnumberssuchthata (cid:17) n b " 1 and d ! 0. Then nP n n i=1 i n (1=a ) n b d ! 0. n i=1 i i P ThisresultisaspecialcaseofPetrov(1995), Lemma6.10. Ifweletb = A andd = A = n A ,then n n n n i=1 i assumption (A-2)guarantees thata " 1andd ! 0,soapplyLemma4toget n n 1 Xn A 2 P P i ! 0: (26) n i=1 A i i=1 i j=1 A j Thestandard ruleforconditional variancegives P n A 2E[V[U jX]] V[L ]−V[E[L jX]] = E[V[L jX]] = i=1Pi i : n n n ( n A )2 i=1 i E[V[U jX]] must be less than one under the actuarial paradigm. Under the mark-to-market paradigm, we i have E[V[U jX]] < E[(cid:7)(X)] < 1 byassumption (A-1). Therefore, there exists afinite constant V (cid:3) such i that P n A 2 1 Xn A 2 1 Xn A 2 E[V[L jX]] (cid:20) V (cid:3) Pi=1 i = V (cid:3)P P i (cid:20) V (cid:3)P P i ! 0: n ( n i=1 A i )2 n i=1 A i i=1 n j=1 A j n i=1 A i i=1 i j=1 A j As E[V[L jX]] must be non-negative and is bounded from above by a quantity converging to zero, it too n mustconverge tozero. QED B Proof of Proposition 3 Almost sure convergence implies convergence in probability (see Billingsley 1995, Theorem 25.2), so for allxand(cid:15) > 0, Pr(jL −E[L jx]j < (cid:15)jx)! 1: (27) n n IfF isthecdfofL ,thenequation (27)implies n n F (E[L jx]+(cid:15)jx)−F (E[L jx]−(cid:15)jx) ! 1: n n n n BecauseF isbounded in[0;1], wemusthaveF (E[L jx]+(cid:15)jx) ! 1andF (E[L jx]−(cid:15)jx) ! 0. n n n n n Let S + denote the set of realizations x of X such that E[L jx] is less than or equal to its q th quantile n n 25

value,i.e., S + (cid:17) fx :E[L jx] (cid:20) (cid:11) (E[L jX])g: n n q n Byconstruction, Pr(x 2 S +)(cid:21) q. n Bytheusualrulesforconditional probability, wehave F ((cid:11) (E[L jX])+(cid:15)) = F ((cid:11) (E[L jX])+(cid:15)jX 2S +)Pr(X 2S +) n q n n q n n n + F ((cid:11) (E[L jX])+(cid:15)jX 62S +)Pr(X 62S +) n q n n n (cid:21) F ((cid:11) (E[L jX])+(cid:15)jX 2S +)Pr(X 2S +) n q n n n (cid:21) F ((cid:11) (E[L jX])+(cid:15)jX 2S +)q (28) n q n n Forallx2 S + ,wehave n F ((cid:11) (E[L jX])+(cid:15)jx) (cid:21) F (E[L jx]+(cid:15)jx) ! 1 n q n n n sothedominated convergence theorem (Billingsley 1995,Theorem16.4)impliesthat F ((cid:11) (E[L jX])+(cid:15)jX 2 S +) ! 1; n q n n sofromequation (28)wehave F ((cid:11) (E[L jX])+(cid:15))! [q;1] n q n asrequired. − Theotherhalfoftheprooffollowssimilarly. Define S as n S − (cid:17) fx: E[L jx](cid:21) (cid:11) (E[L jX])g n n q n sothatPr(x 2 S −)(cid:21) 1−q. Then n F ((cid:11) (E[L jX])−(cid:15)) = F ((cid:11) (E[L jX])−(cid:15)jX 62S −)Pr(X 62 S −) n q n n q n n n + F ((cid:11) (E[L jX])−(cid:15)jX 2 S −)Pr(X 2 S −) n q n n n (cid:20) q+F ((cid:11) (E[L jX])−(cid:15)jX 2 S −)Pr(X 2 S −): (29) n q n n n Forallx2 S − ,wehave n F ((cid:11) (E[L jX])−(cid:15)jx) (cid:20) F (E[L jx]−(cid:15)jx) ! 0 n q n n n 26

sothedominated convergence theorem impliesthat F ((cid:11) (E[L jX])−(cid:15)jX 2 S −) ! 0; n q n n sofromequation (29)wehave F ((cid:11) (E[L jX])−(cid:15))! [0;q] n q n asrequired. QED C Proof of Proposition 5 TheproofofProposition 5requires thefollowinglemma: Lemma5 LetY1andY2berandomvariableswithcdfsF1andF2,respectively.Forallyandall(cid:15) > 0, jF1 (y)−F2 (y)j (cid:20) Pr(jY1 −Y2 j> (cid:15))+maxfF2 (y+(cid:15))−F2 (y);F2 (y)−F2 (y−(cid:15))g: Proof: Corollary ofPetrov(1995,Lemma1.8). To apply Lemma 5, let Y1 = L n with cdf F n and Y2 = E[L n jX] with cdf F n (cid:3) . Fix an open set B and real numbers n0 and (cid:14);(cid:14) for which assumptions (A-4) and (A-5) are satisfied. At every point x^ 2 B, we have jF (E[L jx^])−F (cid:3)(E[L jx^])j (cid:20) Pr(jL −E[L jX]j >(cid:15)) n n n n n n +maxfF (cid:3)(E[L jx^]+(cid:15))−F (cid:3)(E[L jx^]);F (cid:3)(E[L jx^])−F (cid:3)(E[L jx^]−(cid:15))g (30) n n n n n n n n forany(cid:15) >0. Forn > n0,(A-5)guaranteesthatM n (x)isstrictlyincreasingonB,soforallx 2 B,M n (X) (cid:20) M n (x) ifandonlyifX (cid:20) x,soF (cid:3)(M (x)) = H(x)whereH isthecdfofX. n n Fix (cid:15) (cid:3) > 0 such that (x^−(cid:15) (cid:3) ;x^+(cid:15) (cid:3)) (cid:26) B. For any positive (cid:15) < (cid:14)(cid:15) (cid:3) , we then have x^+(cid:15)=(cid:14) 2 B, so M n (x^+(cid:15)=(cid:14))−M n (x^)> (cid:15)foralln > n0. AsF n (cid:3) isnondecreasing, thisimpliesthat F (cid:3)(E[L jx^]+(cid:15))(cid:20) F (cid:3)(M (x^+(cid:15)=(cid:14))) = H(x^+(cid:15)=(cid:14)) n n n n Similarly,wehave F (cid:3)(E[L jx^]−(cid:15)) (cid:21) F (cid:3)(M (x^−(cid:15)=(cid:14))) = H(x^−(cid:15)=(cid:14)): n n n n 27

Thus,foralln> n0, maxfF (cid:3)(E[L jx^]+(cid:15))−F (cid:3)(E[L jx^]);F (cid:3)(E[L jx^])−F (cid:3)(E[L jx^]−(cid:15))g n n n n n n n n (cid:20) maxfH(x^+(cid:15)=(cid:14))−H(x^);H(x^)−H(x^−(cid:15)=(cid:14))g: Assumption (A-5) also provides that H is continuous and increasing on B, so for any (cid:17) > 0 there exists (cid:15) > 0suchthat maxfH(x^+(cid:15)=(cid:14))−H(x^);H(x^)−H(x^−(cid:15)=(cid:14))g < (cid:17): (31) By Proposition 1 and the dominated convergence theorem, L − E[L jX] converges to zero almost n n surely, which implies convergence in probability as well. Therefore, for any choice of (cid:15) > 0 and (cid:17) > 0, thereexistsn < 1suchthat (cid:15) Pr(jL −E[L jX]j >(cid:15)) < (cid:17) 8n> n : (32) n n (cid:15) Combining theseresults, wehavethatforany (cid:17) > 0,there existsan(cid:15) > 0suchthatequations (31)and (32)aresimultaneously satisfiedforn> maxfn0;n (cid:15) g. Thus, lim jF (M (x^))−F (cid:3)(M (x^))j ! 0: (33) n!1 n n n n Setting x^ = (cid:11) (X) and observing that F (cid:3)(M ((cid:11) (X))) = H((cid:11) (X)) = q establishes the first result of q n n q q Proposition 5. For any positive (cid:17) < (cid:14)(cid:15) (cid:3) and n > n0, (A-5) implies that M n ((cid:11) q (X)) −M n ((cid:11) q (X)− (cid:17)=(cid:14)) (cid:20) (cid:17) so F (M ((cid:11) (X))−(cid:17)) (cid:20) F (M ((cid:11) (X)−(cid:17)=(cid:14))). Because (cid:11) (X)−(cid:17)=(cid:14) 2 B, we have by equation (33) n n q n n q q that jF (M ((cid:11) (X)−(cid:17)=(cid:14)))−F (cid:3)(M ((cid:11) (X)−(cid:17)=(cid:14)))j ! 0: n n q n n q For n > n0, F n (cid:3)(M n ((cid:11) q (X) − (cid:17)=(cid:14))) = H((cid:11) q (X) − (cid:17)=(cid:14)) < q, so there exists n~ − < 1 such that F n (M n ((cid:11) q (X) − (cid:17)=(cid:14))) < q for all n > n~ −. Thus, for all n > n~ −, F n (M n ((cid:11) q (X)) − (cid:17)) < q, which impliesthatM ((cid:11) (X))−(cid:17) <(cid:11) (L ). n q q n By a parallel argument, for all positive (cid:17) < (cid:14)(cid:15) (cid:3) there exists n~ + < 1 such that M n ((cid:11) q (X)) − (cid:17) > (cid:11) q (L n )foralln > n~ +. Thus, forall n > maxfn~ −;n~ + g, wehave j(cid:11) q (L n )−M n ((cid:11) q (X))j < (cid:17). As(cid:17) can bemadearbitrarily closetozero,thesecondresultofProposition 5isestablished. QED D Construction of the comparable homogeneous portfolio Momentrestrictionsprovideaconvenientandintuitivewaytomaptheheterogeneousportfoliointoahomogeneous portfolio ofn (cid:3) equal-sized facilities withcommonPDp(cid:22)(cid:3) ,factorloading w (cid:3) ,andLGDparameters 28

(cid:3) (cid:3) (cid:21) and(cid:17) . Lets denotetheshareoftotalportfolio exposureheldinbucket b,i.e., b P A s (cid:17) Pi2b i : b A i i Thefirsttworestrictions equateexposure-weighted expected defaultrateandexpectedportfolio lossrate: XB XB p(cid:22)(cid:3) = p(cid:22) s and (cid:21) (cid:3) p(cid:22)(cid:3) = (cid:21) p(cid:22) s : (34) b b b b b b=1 b=1 (cid:3) Thus,(cid:21) istheexpectedlossratedividedbytheexpecteddefault rate;i.e., P B (cid:21) p(cid:22) s (cid:21) (cid:3) = Pb=1 b b b : (35) B p(cid:22) s b=1 b b Theremaining momentrestrictions equate across the actual and comparable portfolios thecontribution to loss variance from different sources of risk. The contribution of systematic risk (i.e., V[E[LjX]]) takes thesimpleform ! 2 XB V[E[L jX]] = (cid:27) 2 (cid:21) p(cid:22) w s n b b b b b=1 V[E[L (cid:3)jX]] = (cid:27) 2((cid:21) (cid:3) p(cid:22)(cid:3) w (cid:3))2 ; whichimplies P B (cid:21) p(cid:22) w s w (cid:3) = Pb=1 b b b b : (36) B (cid:21) p(cid:22) s b=1 b b b (cid:3) Notethat w issimplyanexpected-loss-weighted averageofthew . b Thecontribution ofidiosyncratic risktolossvariance(i.e.,E[V[L jX]])worksoutto n XB (cid:0) (cid:1) E[V[L jX]] = (cid:21) 2(p(cid:22) (1−p(cid:22) )−(p(cid:22) w (cid:27))2)+p(cid:22) (cid:17) 2 H s 2 n b b b b b b b b b b=1 (cid:16) (cid:17) 1 E[V[L (cid:3)jX]] = (cid:21) (cid:3)2(p(cid:22)(cid:3)(1−p(cid:22)(cid:3))−(p(cid:22)(cid:3) w (cid:3) (cid:27))2)+p(cid:22)(cid:3) (cid:17) (cid:3)2 : n(cid:3) Termscontaining (cid:21) 2(p(cid:22)(1−p(cid:22))−(p(cid:22)w(cid:27))2)represent thecontribution ofidiosyncratic defaultrisk,andterms containing p(cid:22)(cid:17) 2 represent the contribution of idiosyncratic recovery risk. By matching these two contributions separately, I get the final two restrictions needed for identification. The number of exposures in the 29

comparable portfolio worksoutto ! −1 XB n (cid:3) = (cid:3) H s 2 (37) b b b b=1 where (cid:21) 2(p(cid:22) (1−p(cid:22) )−(p(cid:22) w (cid:27))2) (cid:3) (cid:17) b b b b b : b (cid:21)(cid:3)2(p(cid:22)(cid:3)(1−p(cid:22)(cid:3))−(p(cid:22)(cid:3)w(cid:3)(cid:27))2) Finally,thevarianceofLGDforthecomparable portfolio isgivenby (cid:3) XB n (cid:17) (cid:3)2 = (cid:17) 2 p(cid:22) H s 2 : (38) p(cid:22)(cid:3) b b b b b=1 E Proof of Proposition 6 ExpectedshortfallisthesumofexpectedlossinthetailandacorrectiontermformassattheVaRboundary. Under the given assumptions, the latter term disappears asymptotically both in ES [L ]and ES [M (X)]. q n q n Forn > n0, the variable M n (X) iscontinuous in the neighborhood ofM n ((cid:11) q (X)) and, by Proposition 4, (cid:11) (M (X)) = M ((cid:11) (X));thisimpliesPr(M (X) <M ((cid:11) (X))) = q. ByChebyshev’sinequality and q n n q n n q assumption (A-1)wehave E[M (X)2] E[(cid:7)(X)] M ((cid:11) (X))2 (cid:20) n (cid:20) < 1; n q Pr(M (X) > M ((cid:11) (X))) 1−q n n q sothesequence fM ((cid:11) (X))gisbounded fromabove. Therefore, n q (cid:11) q (M n (X))(q −Pr(M n (X) < (cid:11) q (M n (X)))) = 0 8n > n0: Although L need not be continuous, arguments parallel to those used in the proof of Proposition 5 n showthatPr(L (cid:21) (cid:11) (L )) ! 1−q. Thatproposition alsoprovidesthatj(cid:11)(L )−M ((cid:11) (X))j ! 0,so n q n n n q (cid:11) (L )isasymptotically bounded fromabove. Thisimplies q n lim j(cid:11) (L )(q−Pr(L < (cid:11) (L )))j = 0: q n n q n n!1 Tocompletetheproof, wenowonlyneedshowthat jE[L n (cid:1)1 fLn (cid:21)(cid:11)q (Ln )g ]−E[M n (X)(cid:1)1 fMn (X)(cid:21)(cid:11)q (Mn (X))g ]j ! 0: (39) Let Y (cid:17) (L − (cid:11) (L )) and let Y ^ (cid:17) (M (X) − M ((cid:11) (X))). The terms of equation (39) can be n n q n n n n q 30

re-writtenas E[M n (X)(cid:1)1 fMn (X)(cid:21)(cid:11)q (Mn (X))g ] = E[(M n (X)−M n ((cid:11) q (X)))(cid:1)1 fMn (X)−Mn ((cid:11)q (X))(cid:21)0g ]+M n ((cid:11) q (X))E[1 fMn (X)(cid:21)Mn ((cid:11)q (X))g ] =E[maxfY ^ ;0g]+M ((cid:11) (X))Pr(M (X) (cid:21) M ((cid:11) (X))) n n q n n q andsimilarly E[L n (cid:1)1 fLn (cid:21)(cid:11)q (Ln )g ] = E[maxfY n ;0g]+(cid:11) q (L n )Pr(L n (cid:21) (cid:11) q (L n )): Asj(cid:11) (L )−M ((cid:11) (X))j ! 0and q n n q lim Pr(L (cid:21) (cid:11) (L )) = lim Pr(M (X) (cid:21) M ((cid:11) (X))) = 1−q; n q n n n q n!1 n!1 wehave j(cid:11) (L )Pr(L (cid:21) (cid:11) (L ))−M ((cid:11) (X))Pr(M (X) (cid:21) M ((cid:11) (X)))j ! 0: q n n q n n q n n q Forally;y^2<,jmax(y;0)−max(y^;0)j (cid:20) jy−y^j. Therefore, jE[maxfY ;0g]−E[maxfY ^ ;0g]j (cid:20) E[jY −Y ^ j] (cid:20) E[jL −M (X)j]+j(cid:11) (L )−M ((cid:11) (X))j: n n n n n n q n n q Aseachofthesetermsconverges tozero,equation (39)isestablished. QED F Proof of Lemma 1 DivideΩintotwosubsets B1 = f! : 0(cid:20)min(Y1 (!);Y2 (!))_max(Y1 (!);Y2 (!))(cid:20)0g B2 = f! : (Y1 (!)<0<Y2 (!))_(Y2 (!)<0<Y1 (!))g: Observe that B1 [B2 = Ω and B1 \B2 = ;. IfY isan integrable random variable on (Ω;F;P), wecan write Z E[Y +] = max(Y(!);0)P(d!) ZΩ Z = max(Y(!);0)P(d!)+ max(Y(!);0)P(d!): B1 B2 31

The set B1 contains all ! for which Y1 and Y2 are either both positive or both negative. Under both these circumstances, max(Y1 (!)+Y2 (!);0)equals max(Y1 (!);0)+max(Y2 (!);0),so Z max(Y1 (!)+Y2 (!);0)P(d!) B1 Z Z = max(Y1 (!);0)P(d!)+ max(Y2 (!);0)P(d!): (40) B1 B1 ThesetB2 contains all! forwhichY1 andY2 areofoppositesign,so Z max(Y1 (!)+Y2 (!);0)P(d!) B2 Z Z (cid:20) max(Y1 (!);0)P(d!)+ max(Y2 (!);0)P(d!): (41) B2 B2 Summingleftandrighthandsidesofequations (40)and(41),weobtain E[(Y1 +Y2 )+] (cid:20) E[Y 1 +]+E[Y 2 +]: (42) If P(B2 ) > 0, then the inequality in equation (41) isstrict, and therefore the inequality inequation (42) is strictaswell. + G Asymptotic EEL in CreditRisk I derive the asymptotic EEL for a homogeneous portfolio under a single systematic factor version of CreditRisk + . Let p(cid:22)denote default probability, (cid:21) denote LGD, w denote factor loading, and (cid:27) denote the + volatilityofsystematicfactorX. TheconditionalexpectedlossrateintheCreditRisk specificationisgiven byequation(17). Asn! 1,L converges to(cid:22)(X),soasymptotic EELisequaltothevalueofcsolving n Z 1 (cid:18) = E[((cid:22)(X)−c)+]= ((cid:22)(x)−c)h(x)dx; (43) (cid:22)−1(c) where h((cid:1)) is the gamma pdf with mean one, variance (cid:27) 2 . Using Abramowitz and Stegun (1968, 6.5.1, 6.5.21)tosolvethisintegral, Iobtain (cid:18) = (EL−c)(1−H((cid:22) −1(c)))+ EL(cid:1)w (cid:0) (cid:22) −1(c)=(cid:27) 2 (cid:1) 1=(cid:27)2 exp (cid:0) −(cid:22) −1(c)=(cid:27) 2 (cid:1) ; (44) Γ(1+1=(cid:27)2) whereH((cid:1))denotes thegammacdf,ELisexpected loss((cid:21)p(cid:22)),and c−(1−w)(cid:1)EL (cid:22) −1(c) = : w(cid:1)EL The gamma cdf is available in nearly all numerical packages. Standard software for solving nonlinear equations quickly finds the capital ratio c which covers EEL target (cid:18). In the special case of (cid:27) = 1, the 32

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